Collaborative Research: Algebraic Geometry of Tensors

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。pi将研究传统研究品种的高割线品种，如Segre品种、Grassmann品种和Segre- veronese品种。这些变化对应于秩一张量、交替张量、正则张量和对称张量的杂化的参数空间，它们的（闭）高割线变化对应于高秩张量参数空间的紧化。研究的主要目的是对Segre品种、Grassmann品种和Segre- veronese品种的割线缺陷品种进行分类。这类似于著名的Alexander和Hirschowitz的定理，该定理断言Veronese变体的高割线变体具有预期的维数（对完全描述的例外列表取模）。这项工作完成了多项式的韦林问题，这个问题在一段时间内一直是一个悬而未决的问题。对于Segre品种和Grassmann品种，有一个相应的、推测的完整的割线缺陷品种列表。该项目的第一个组成部分是对现有方法的改进，并开发新的理论和算法方法来解决这个分类问题。项目的第二个部分是关于张量的分解。在许多应用程序中，将数据集合表示为多索引列表是很自然的。或者，可以将数据视为多维数组（有时称为多路数组）。例如，数字灰度图像可以存储为数字矩阵，其中图像中的每个像素位置对应于矩阵中的位置，矩阵中的数字对应于像素的暗度。以类似的方式，数字彩色图像可以存储为三维数字数组。一个数学框架，包括研究多向数组，以及它们作为更基本对象的和的表示，是通过张量的参数空间。本课题从代数几何点的角度探讨与张量、张量分解、张量秩和张量边界秩相关的问题。这些学科在信号处理、数据分析、计算生物学、组合学、代数几何和统计学等领域都有重要的应用。因此，期望通过这项研究开发的技术将促进我们对多个学科的知识和理解。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PIs will study higher secant varieties of classically studied varieties such as Segre varieties, Grassmann varieties, and Segre-Veronese varieties. These varieties correspond to parameter spaces for rank one tensors, alternating tensors, and hybrids of regular tensors and symmetric tensors, and their (closed) higher secant varieties correspond to compactifications of the parameter spaces for higher rank tensors. The main goal of the research is the classification of defective secant varieties of Segre varieties, Grassmann varieties and Segre-Veronese varieties. This is analogous to the celebrated theorem of Alexander and Hirschowitz, which asserts that higher secant varieties of Veronese varieties have the expected dimension (modulo a fully described list of exceptions). This work completed the Waring problem for polynomials which had stood for some time as an outstanding unsolved problem. There is a corresponding, conjectural complete list of defective secant varieties for Segre varieties and for Grassmann varieties. The first component of the project is on the refinement of existing methods and the development of new theoretical and algorithmic methods towards the solution of this classification problem. The second component of the project is concerned with decomposition of tensors.In many applications, it is natural to represent a collection of data as a multi-indexed list. Alternatively, one can think of the data as a multidimensional array (sometimes called a multi-way array). For example, a digital grayscale picture can be stored as a matrix of numbers where each pixel location in the picture corresponds to a location in the matrix and the number in the matrix corresponds to the darkness of the pixel. In a similar manner, a digital color picture can be stored as a three dimensional array of numbers. A mathematical framework that includes the study of multi-way arrays, and their representations as sums of more basic objects, is through parameter spaces of tensors. This project explores problems related to tensors, tensor decomposition, tensor rank and tensor border rank from an algebro-geometric point viewpoint. These subjects have significant applications in fields as diverse as signal processing, data analysis, computational biology, combinatorics, algebraic geometry and statistics. It is the expectation, therefore, that techniques developed through this research will advance our knowledge and understanding across multiple disciplines.